Substochastic semigroups for transport equations with conservative boundary conditions

We consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a C₀-semigroup in L¹. With techniques borrowed from the additive perturbation theory of substochastic semigroups, we describe precisely its domain and provide necessary and sufficient conditions ensuring to be stochastic. We apply these results to examples from kinetic theory.